
and no group element \(\hat{\gamma }\in Sp(2h^{2,1}+2, \mathbb {Z})\)  exists such that \({\cal Q}(\gamma )=\hat{\gamma }^{-1}\gamma \hat{\gamma }\) .
In this respect, we argue that CP has a geometrical origin in the CY moduli space,
namely the outer automorphism of the \(Sp(2h^{2,1}+2, \mathbb {Z})\)  modular group.
The whole group is described by the semidirect product group \(Sp(2h^{2,1}+2, \mathbb {Z}) \rtimes {\cal CP}\) , since there exists a group homomorphism from \({\cal CP}\)  to the automorphism group of \(Sp(2h^{2,1}+2, \mathbb {Z})\)  and \(Sp(2h^{2,1}+2, \mathbb {Z})\)  is a normal subgroup of \(Sp(2h^{2,1}+2, \mathbb {Z}) \rtimes {\cal CP}\) .
That is a natural extension of the \(SL(2,\mathbb {Z})\simeq Sp(2, \mathbb {Z})\)  modular
group known in the \(T^2\)  toroidal background [1]}, [2]},
where the whole group is given by the “generalized modular group"
\(GL(2,\mathbb {Z})\simeq SL(2, \mathbb {Z}) \rtimes {\cal CP}\) .We can treat an element of \(Sp(2 h^{2,1}+2, \mathbb {Z}) \rtimes {\cal CP}\)  as a pair \((g, h)\)  where \(g \in Sp(2 h^{2,1}+2, \mathbb {Z})\) , \(h \in {\cal CP} \simeq \mathbb {Z}_2\) , and a product between two pairs is represented as \((g_1, h_1) (g_2, h_2 ) = (g_1 h_1^{-1} g_2 h_1, h_1 h_2)\)  with respect to the outer automorphism. Nevertheless, in the toroidal case, it is enough to consider a set of elements \((g, h) (h \ne 1)\)  as a negative determinant part of \(GL(2, \mathbb {Z})\) , so we simply use \(GL(2, \mathbb {Z})\)  instead of \(SL(2, \mathbb {Z}) \rtimes {\cal CP}\)  thanks to the isomorphism. However, because of the explicit form of \(\hat{\sigma }^3\) , we cannot think \(Sp(2 h^{2,1}+2, \mathbb {Z}) \rtimes {\cal CP}\)  as the same with the toroidal case for odd \(h^{2,1}\) , where \({\cal CP}\)  has a positive determinant. For even \(h^{2,1}\) , \({\cal CP}\)  has a negative determinant then it could be similar to the toroidal case that we can distinguish new elements of the whole group \(Sp(2 h^{2,1}+2, \mathbb {Z}) \rtimes {\cal CP}\)  that are not in original \(Sp(2 h^{2,1} +2, \mathbb {Z})\)  by their signs of determinants.
